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Physical Review Letters

Critical Exponents for Charged and Neutral Bose Gases above λ Points

Shang-keng Ma

Published Nov 6, 1972• 92 citations
Publisher

Abstract

Critical behavior and exponents above the $\ensuremath{\lambda}$-transition points in charged and neutral Bose systems are examined with emphasis on the correlation of density fluctuations. Exponents are calculated with Wilson's expansion methods for arbitrary $d \mathrm{to} O(\frac{1}{m})$, where $d$ is the dimension and $m$ the number of components of the Bose field. Some results for arbitrary $m \mathrm{to} O({\ensuremath{\epsilon}}^{2})(\ensuremath{\epsilon}=4\ensuremath{-}d)$ are also obtained.

DOI
10.1103/physrevlett.29.1311

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Critical behavior and exponents above the \ensuremathλ\ensuremath{\lambda}\ensuremathλ-transition points in charged and neutral Bose systems are examined with emphasis on the correlation of density fluctuations. Exponents are calculated with Wilson's expansion methods for arbitrary dtoO(1m)d \mathrm{to} O(\frac{1}{m})dtoO(m1​), where ddd is the dimension and mmm the number of components of the Bose field. Some results for arbitrary mtoO(\ensuremathϵ2)(\ensuremathϵ=4\ensuremath−d)m \mathrm{to} O({\ensuremath{\epsilon}}^{2})(\ensuremath{\epsilon}=4\ensuremath{-}d)mtoO(\ensuremathϵ2)(\ensuremathϵ=4\ensuremath−d) are also obtained.
In earlier work, Ma [S.K. MA, Phys. Rev. Lett., 29, 1311 (1972)] has studied the critical exponents γ and η for charged and neutral Bose gases. Here we use the result of Ma, valid for general dimensionality d but only to O(m −1), where m is the number of components of the Bose field, to write a relation between γ(d) and η(d) to O(m −1). This then motivates, but now for the Ising model, a relationship between the critical exponents γ and η, via the dimensionality d. We finally demonstrate a connection between the two renormalisation group eigenvalues y t and y h , via the critical exponent δ with a dimensional dependence.
Summary form only given. Phase transitions are among the most dramatic phenomena in nature. Minute variations in the conditions controlling a system trigger a fundamental change of its properties. In the critical regime near a second-order phase transition point the fluctuations of the order parameter are so dominant that they completely govern the behaviour of the system on all length scales. The correlation length of the ultracold gas measures the characteristic length scale of these fluctuations and diverges at the critical temperature. Using cold atomic gases, we have directly observed critical fluctuations of the order parameter near the phase transition of Bose-Einstein condensation. From the divergence of the correlation length of the order parameter versus temperature we have determined its critical exponent v. Despite our densities being ten orders of magnitude smaller and our interactions considerably weaker, we find a remarkable similarity to case of the lambda-transition of helium-4.
We study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, characterizing the universal behaviors of the critical modes arising at the BEC transitions. For this purpose, we use field-theoretical (FT) renormalization-group (RG) methods and perform mean-field and numerical calculations. The FT RG analysis is based on the Landau-Ginzburg-Wilson ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory with two complex scalar fields which has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z2{\mathbb{Z}}_{2}Z2​ symmetry, coupled by effective density-density interactions, the global symmetry is ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$. At the BEC transition, it may break into ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when both components condense simultaneously, or to $\mathrm{U}(1)\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when only one component condenses. This implies different universality classes for the corresponding critical behaviors. Numerical simulations of the two-component Bose-Hubbard model in the hard-core limit support the RG prediction: when both components condense simultaneously, the critical behavior is controlled by a decoupled XY fixed point, with unusual slowly decaying scaling corrections arising from the onsite interspecies interaction.
Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order O\mathcal{O}O, O\mathcal{O}O phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (O=2\mathcal{O}=2O=2) and Gaussian tricritical points (O=3\mathcal{O}=3O=3) to critical points of arbitrary order O\mathcal{O}O. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for O=3,4\mathcal{O}=3,4O=3,4 are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for O=3\mathcal{O}=3O=3, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.
We develop a practical theoretical formalism for studying the critical properties of a trapped Bose-Einstein condensate using the projected Gross-Pitaevskii equation. We show that this approach allows us to investigate the behavior of the correlation length, condensate mode, and its number fluctuations about the critical point. Motivated by recent experiments [Donner et al., Science 315, 1556 (2007)], we calculate the critical exponent for the correlation length of the trapped system. We observe finite-size effects in our results when the correlation length becomes comparable to the Ginzburg length. We extend the Binder cumulant to the trapped system and discuss an experimental method for measuring number fluctuations.
We analyze density-density correlations of expanding clouds of weakly interacting two-dimensional Bose gases below and above the Berezinskii-Kosterlitz-Thouless transition, with particular focus on short-time expansions. During time-of-flight expansion, phase fluctuations of the trapped system translate into density fluctuations, in addition to the density fluctuations that exist in situ. We calculate the correlations of these fluctuations both in real space and in momentum space and derive analytic expressions in momentum space. Below the transition, the correlation functions show an oscillatory behavior, controlled by the scaling exponent of the quasicondensed phase, due to constructive interference. We argue that this can be used to extract the scaling exponent of the quasicondensate experimentally. Above the transition, the interference is rapidly suppressed when the atoms travel an average distance beyond the correlation length. This can be used to distinguish the two phases qualitatively.